VGAM (version 0.7-8)

invbinomial: Inverse Binomial Distribution Family Function

Description

Estimates the two parameters of an inverse binomial distribution by maximum likelihood estimation.

Usage

invbinomial(lrho="elogit", llambda="loge",
            erho=if(lrho=="elogit") list(min = 0.5, max = 1) else list(),
            elambda=list(), irho=NULL, ilambda=NULL, zero=NULL)

Arguments

lrho, llambda

Link function for the $\rho$ and $\lambda$ parameters. See Links for more choices.

erho, elambda

List. Extra argument for each of the links. See earg in Links for general information.

irho, ilambda

Numeric. Optional initial values for $\rho$ and $\lambda$.

zero

See CommonVGAMffArguments.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Details

The inverse binomial distribution of Yanagimoto (1989) has density function $$f(y;\rho,\lambda) = \frac{ \lambda \,\Gamma(2y+\lambda) }{\Gamma(y+1) \, \Gamma(y+\lambda+1) } { \rho(1-\rho) }^y \rho^{\lambda}$$ where $y=0,1,2,\ldots$ and $\frac12 < \rho < 1$, and $\lambda > 0$. The first two moments exist for $\rho>\frac12$; then the mean is $\lambda (1-\rho) /(2 \rho-1)$ (returned as the fitted values) and the variance is $\lambda \rho (1-\rho) /(2 \rho-1)^3$. The inverse binomial distribution is a special case of the generalized negative binomial distribution of Jain and Consul (1971). It holds that $Var(Y) > E(Y)$ so that the inverse binomial distribution is overdispersed compared with the Poisson distribution.

References

Yanagimoto, T. (1989) The inverse binomial distribution as a statistical model. Communications in Statistics: Theory and Methods, 18, 3625--3633.

Jain, G. C. and Consul, P. C. (1971) A generalized negative binomial distribution. SIAM Journal on Applied Mathematics, 21, 501--513.

Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman & Hall

See Also

negbinomial, poissonff.

Examples

y <- rnbinom(n <- 1000, mu=exp(3), size=exp(1))
fit  <- vglm(y ~ 1, invbinomial, trace=TRUE)
c(mean(y), fitted(fit)[1:5])
summary(fit)
coef(fit, matrix=TRUE)
Coef(fit)
sum(weights(fit))  # sum of the prior weights
sum(weights(fit, type="w")) # sum of the working weights